Multiple prehension manipulator

ABSTRACT

A multiple prehension manipulator mechanism including a base; a plurality of finger assemblies mounted on the base; finger drive means for selectively opening and closing the fingers so that each finger moves in a single curling plane; and positioning drive means for selectively positioning the finger assemblies so that different prehensile modes can be achieved. The disclosure also contemplates the method of operation of the mechanism.

United States Patent [191 Skinner, II

[ 51 Feb. 18, 1975 MULTIPLE PREHENSION MANIPULATOR [76] Inventor: FrankR. Skinner, II, 2248 Ann Dr.,

St. Joseph, Mich.

[22] Filed: Mar. 5, 1974 211 Appl. No.: 448,342

Related US. Application Data [62] Division of Ser. No. 360,022, May 14,1973,

abandoned.

[52] U.S. Cl. 294/106, 3/l2.7, 214/1 CM [51] Int. Cl. A6lf 1/06 [58]Field of Search 214/1 CM; 3/l2.7; 294/97, 294/115, 106

[56] References Cited UNITED STATES PATENTS Colechia 214/1 CM 3,694,0219/ l 972 Mullen 294/106 Primary Examiner-Robert J. Spar AssistantExaminerGe0rge F. Abraham Attorney, Agent, or FirmB. J. Powell [5 7]ABSTRACT 9 Claims, 5 Drawing Figures MULTIPLE PREHENSION MANIPULATORCROSS-REFERENCE TO RELATED APPLICATIONS This application is a divisionof my copending application Ser. No. 360,022 filed May 14, 1973 forMultiple Prehension Manipulator Mechanism,-

now abandoned.

BACKGROUND OF THE INVENTION Many attempts known been made to produce amanipulator having substantially the same capabilities as the humanhand. Because the human hand has many motor and control systems, suchprior art manipulators have been very complicated and thereforeprohibitively expensive to manufacture and maintain. Because of thecomplexity of the human hand, many of these prior art manipulatorsattempted to combine several motor functions of the human hand with theattendant loss of capability.

SUMMARY OF THE INVENTION The invention disclosed herein overcomes theseand other problems associated with the prior art by providing amanipulator which has virtually all ofthe basic capabilities associatedwith the human hand. The construction of the invention is relativelysimple thereby reducing the manufacturing cost and maintenance cost.

The invention comprises generally a plurality of finger assemblies, eachincluding a finger pivoted about at least one finger axis through asingle plane normal to the finger axis, a base mounting the fingerassemblies so that the plane of each finger can be rotated about apositioning axis through the plane and normal to the finger axes, fingerdrive means for pivoting the fingers about the respective finger axes,and positioning means for moving the finger assemblies so that at leasttwo of the planes will be rotated about their respective positioningaxes. The finger drive means may individually or collectively pivot thefingers about their finger axes. Also, the planes of movement of thefingers may be rotated so that the positioning axis substantiallyintersects the finger axes.

These and other features and advantages will become more apparent uponconsideration of the following specification and accompanying drawingswherein like characters of reference designate corresponding partsthroughout the various views and in which:

BRIEF DESCRIPTION OF THE DRAWINGS FIG. I is a perspective view of oneembodiment of the invention;

FIG. 2 is a side view of the manipulator of FIG. 1 shown partly incross-section with the fingers in tip prehensile mode and the fingerdrive mechanism omitted for clarity;

FIG. 3 is a cross-sectional view taken along line 33 in FIG. 2 andshowing the finger assemblies in threejaw prehensile mode;

FIG. 4 is an operating end view of the manipulator of FIG. 1; and,

FIG. 5 is a cross-sectional view of the finger mechanism taken alongline 5-5 in FIG. 4.

These figures and the following detailed description disclose specificembodiments of the invention, however, it is to be understood that theinventive concept is not limited thereto since it may be embodied inother forms.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS Referring to FIGS. l5,it will be seen that the first embodiment of the manipulator isdesignated by the numeral 10. Generally, the manipulator 10 includes abase 11, a plurality of finger assemblies 12 mounted on base 11, afinger driving mechanism 14 carried by each finger assembly 12, and apositioning mechanism 15 carried in base 11 for positioning the fingerassemblies 12.

Basically, the manipulator is an assembly of drives and mechanismsintended for prehension. These mechanisms, called fingers, can have oneor more bending sections. Externally, a finger with its bending sectionsresembles an open linkage. Each finger link is a component of a closedlinkage which can pivot or rotate the link. The fingers do not translateand are attached to a base. Three fingers are considered necessary andsufficient in the construction of the manipulator. Fingers can approach,contact, or pass one another during prehensile operation. Themanipulator contains all of the drives either in the base or fingerassemblies. A multiple degree-of-freedom wrist mechanism (not shown)'may be used to connect to and move the manipulator so that it canapproach an object from any direction The objective of the manipulatoris to produce a highly versatile hand with a minimum number of movingparts, a dependable drive system, and an optimum number of degrees offreedom. The number of degrees of freedom is considered optimum when itis estimated that the manipulator can grasp all of the basic geometricalshapes from any aspect with the minimum number of external controlinputs. These basic shapes are rectangular and triangular prisms,spheres, and cylinders.

The human hand is generally accepted as being capable of the six basicprehensile patterns: lateral, hook, tip, palmar, spherical, andcylindrical. These six basic prehensile patterns. because ofsimilarities, can be reduced to the three basic mechanical equivalentsof wrap, three-jaw and tip prehension that very nearly duplicate thebasic human hand prehensile patterns. Additionally, in order to griplarge objects that the fingers can not surround, spread prehension iscreated which consists of inserting the fingers into an opening in theobject and then bending them outward to engage the object within theopening. The manipulator 10 is able to generate all of the above fourequivalent prehensile ipatterns.

Referring now to FIGS. 24, it will be seen that the base 11 is hollowwith an equilateral triangular shaped top palmar plate 21 and bottombase plate 22. A tubular side wall 24 also having a general equilateraltriangular cross-sectional shape connects the plates 21 and 22 so thatthe plates are substantially parallel to each other and in verticalalignment. Generally speaking, the

palmar plate 21 corresponds to the palm of the human hand. A cavity 25is defined within the plates 21 and 22 and side wall 24. requiresvelocity A finger assembly 12 is rotatably journalled between plates 21and 22 at each of their three corners and extends outwardly from thepalm plate 21. The finger asseniblies 12 are individually designated12., -l2 to distinguish each. The finger driving mechanism 14 of eachassembly 12 is positioned in cavity 25 and rotatable with its associatedassembly 12 as will become more apparent. The positioning mechanism 15is also mounted within cavity 25.

FINGER ASSEMBLY Any of many systems may be used to pivot links Fingerassemblies 12 42, are all identical in con- 5 40-42 with respect to eachother and the finger 12 with struction and therefore only one assemblywhich is designated generally 12 as seen in FIG. 5 will be described indetail with like reference numbers applied to each. The assembly 12includes a frame 31 which is rotatably journalled about a positioningaxis PA between palmar plate 21 and base plate 22 as seen in FIG. 2. Amounting bracket 32 on frame 31 projects through an appropriate opening26 in palmar plate 21. A finger 34 is connected to bracket 32 through asingle revolute joint 35 so the entire finger 34 is pivotal through asingle curling plane CP as best seen in FIG. 1 about a curling axis CApassing through the joint 35. It will be noted that the positioning axisPA is parallel to the curling axis CA passing through the joint 35. Itwill be noted that the positioning axis PA is parallel to the curlingplane CP while the curling axis CA is perpendicular to the curling planeCP. In this embodiment, it will be noted that the positioning axis PAsubstantially intersects the curling axis CA and lies within the curlingplane CP.

While fingers 34 may be a single rigid member, those illustrated havemultiple links joined by revolute joints to increase the ability of thefinger to constrain an object and thus its versatility. The fingers 34may have any number of links, however, three links are illustrated andhave been found sufficient when used in combination with other likefingers to approximate the human hand. The links illustrated are a baselink 40, an intermediate link 41 and a distal link 42. The base link 40is connected at one end to the bracket 32 through the revolute joint 35.One end of the intermediate link 41 is connected to the other end of thebase link 40 through a revolute joint 44 so that link 41 is pivoted tolink 40 about a second curling axis CA-2 parallel to the axis CA ofjoint35. One end of the distal link 42 is connected to the other end of theintermediate link 41 through a third revolute joint 45 so that link 42is pivoted to link 41 about a third curling axis CA-3 parallel to thesecond axis CA-2 and base axis CA.

Each link is provided with a gripping surface 46 to engage an object asis apparent. While only one gripping surface 46 is provided on one sideof each link, it is to be understood that such surfaces may be providedon both sides if the finger is to be double acting as will become moreapparent.

To limit the amount of movement between links 40-42 and with bracket 32,a stop 48 may be provided on each end of base link 40 at the revolutejoints 35 and 44 as well as on the distal link 42 at the revolute joint45. The stops 48 may be configurated differently depending on themovement to be controlled, however,.

the stops 48 illustrated limit the opening movement of the links to aposition in which the links are longitudinally aligned in asubstantially straight path P as seen in FIGS. 2 and 5 and the entirefinger 34 so that the path P is substantially normal to the workingsurface 23 of the palmar plate 21 when the finger assembly 12 is in itsnormal open position.

It will also be understood that while the fingers 34 could be curledaway from path P in either direction in a plane CP, the embodiment showncurls the fingers 34 away from path P in only one direction and that isin the direction toward the surface 46 as indicated by arrow C in FIGS.2 and 5. The finger 12 is shown in a first curled position in FIG. 2 bydashed lines and in anrespect to bracket 32 and the working surface ofpalmar plate 21. For instance, the links 4042 may be pivotedindependently at each revolute joint 35,44 and 45 or the pivoting at thejoints may be coupled in any desired fashion. Several systems which canbe used are cross four-bar chains, miniature compound pulleys, four-barchains with an expanding link and tension cables. Cross fourbar chainsare dependable, easily built, and can transmit an angular displacementto the finger links in a continuously compounding manner. Miniaturecompound pulleys develop a high mechanical advantage and allow thefinger links to bend through large angles. Expanding link four-barchains can be used to drive the links easily with a high mechanicaladvantage and reasonable losses. Tension cables rotate the links bysimple direct contact and require very little space. The finger drivingmechanism 14 is shown by way of example only and is a tensioncable-pulley system.

The mechanism 14 is mounted in frame 31 and rotatable therewith so thatthe relative positions are maintained between the finger 34 and drivingmechanism 14 as the finger assembly 12 is rotated about its position ingaxis PA. Because the motion of the finger 34 is attempting to duplicatethat of the human finger, the base revolute joint 35 is independentlyrotated while the second and third revolute joints 44 and 45 areconcurrently rotated in this illustration.

The mechanism 14 includes a curl drive unit 50 and a pinch drive unit 51as best seen in FIG. 5. Unit 50 includes a reversible drive motor 52drivingly connected to a first multiple sheave winding drum 54. A thirdcable drive is operatively connected to one sheave of drum 54 and servesto selectively close the third revolute joint 45. Third cable drive 55includes a pair of pulleys 56, one being rotatably journalled in thedistal link 42 on the opposite side from the revolute joint 45 and onebeing rotatably journalled in the intermediate link 41 on the oppositeside from joint 45. A flexible cable 58 is wound around pulleys 56 andattached to drum 54 in such a way that when cable 58 is wound upon drum54 as drum 54 is rotated in a first direction, the cable 58 forces thepulleys 56 toward each other to force the link 45 to pivot toward link44 about the third curl axis CA-3. A second cable drive 60 similar todrive 55 is operatively connected to another sheave of the winding drum54 to selectively close the second revolute joint 44. The second drive60 includes a pair of pulleys 61 journalled in links 41 and 40 acrossjoint 44 and drive cable 62 wound therearound and connected to drum 55so that, as the drum rotates in the first direction, the pulleys 61 willbe drawn together to pivot link 41 toward link 40 about the second curlaxis CA-Z.

The unit 50 also includes an uncurling cable drive 64 operativelyconnected to yet another sheave of the first winding drum 54 and servesto concurrently open the .second and third revolute joints 44 and 45.The drive 64 includes a pair of pulleys 65, one rotatably journalled inthe inboard end of link 42 at the stop 48 and the other rotatablyjournalled in the outboard end of base link 40 at stop 48. A slackadjustment pulley 66 is rotatably journalled in link 41 intermediate itsends and is spring urged toward the gripping surface 46. A flexiblecable 68 is wound around pulleys 65 and 66 and connected to winding drum54 in such a manner that when drum 54 is rotated oppositely to the firstdirection, the pulleys 65 will be forced toward pulley 66 to open thelinks 41 and 42 about the curling axes CA-2 and CA-3. Thus, as the drum54 is rotated in the first direction (counterclockwise in FIG. 5), thelinks 41 and 42 will be concurrently curled, and as drum 54 is rotatedin the opposite direction (clockwise in FIG. 5) the links 41 and 42 willbe uncurled to straighten the finger 34.

Unit 51 includes a reversible drive motor 70 drivingly connected to asecond multiple sheave winding drum 71. A first cable drive 72 isoperatively connected to one sheave of drum 71 and serves to selectivelyclose the first revolute joint 35. First cable drive 72 includes a pairof pulleys 74, one being rotatably journalled in the base link 40 on theopposite side from the revolute joint 35 and one being rotatablyjournalled in the bracket 32 on the opposite side from joint 35. Aflexible cable 75 is wound around pulleys 74 and attached to drum 71 insuch a way that when cable 75 is wound upon drum 71, as the drum 71 isrotated in a first direction, the cable 75 forces the pulleys 74 towardeach other to force the link 40 to pivot toward bracket 32 about thefirst curl axis CA to impart a pinching movement to finger 34.

The unit 51 also includes an unpinching cable drive 76 operativelyconnected to another sheave of the second winding drum 71 and serves toconcurrently open the first revolute joint 35. The drive 76 includes apulley 78 rotatably journalled in the inboard end oflink 40 at the stop48. Pulley 79 is rotatably journalled in bracket 32 and is spring urgedaway from revolute joint 35. A flexible cable 80 is wound around pulleys78 and 79 and connected to winding drum 71 in such a manner that whendrum 71 is rotated oppositely to the first direction, the pulley 78 willbe forced toward pulley 79 to open the link 40 about the curling axisCA. Thus, as the drum 71 is rotated in the first direction (clockwise inFIG. 5), the link 40 will be rotated to cause the finger 34 to pinch,and as drum 71 is rotated in the opposite direction (counterclockwise inFIG. 5) the link 40 will be unpinched to straighten the finger 34.

POSITIONING MECHANISM The positioning mechanism uses a single drivemotor to drive the finger assemblies 12 into their various prehensilemodes. If the fingers 34 can be pinched or curled in both directions inthe curling plane CP away from their normal path P, then it is necessaryto rotate only two finger assemblies 12 about their positioning axes PA.While various mechanisms may be used to rotate the assemblies 12, themechanism shown rotates all three finger assemblies 12 to allow thefingers 34 to be pinched or curled in only one direction away fromnormal path P.

Referring now to FIG. 3, the mechanism 15 includes a positioning drivemotor 81 mounted on palmar plate 21 within cavity 25. The drive shaft 82of motor 81 mounts drive gear 84 on the lower end thereof with a pitchdiameter d,. The drive gear 84 meshes directly with a gear 85 on thepivot shaft 38 of finger assembly and drives a driven gear 86 on shaft38 of assembly 12,

through idler gear 88. Gears 85, 86 and 88 have pitch diameters d, d,,and (1., respectively. A four-bar link 89 is pinned at one end to 84 bydrive pin 90 located from the rotational axis of gear 84 a distance r,as will become more apparent. The link 89 has an effective length L andits opposite end is pinned to a four-bar gear 91 through a driven pin92. The pin 92 is located from the rotational axis of gear 91 a distancer: as will become more apparent. The four-bar gear 91 meshes with adriven gear 94 on the pivot shaft 38 of finger assembly 12 to drivesame.

The mechanism 15 is known as a double dwell mechanism and progressivelyrotates finger assemblies 12 from the three-jaw prehensile mode (bendingdirection shown by solid lines in FIGS. 3 and 4) to the wrap prehensilemode (bending direction shown by dashed lines) to the spread prehensilemode (bending direction shown by phantom lines) to the tip prehensilemode (bending direction shown by dotdash lines). As gear 84 is rotated,the drive pin 90 assumes the solid line position on gear 84 in FIG. 3labelled P, for the three-jaw prehensile mode; assumes the dashed lineposition labelled P,, on gear 84 for the wrap prehensile mode which isdisplaced from position P, by angle a; assumes the phantom line positionlabelled P, on gear 84 for the spread prehensile mode which is displacedfrom position P by angle B; and assumes the dot-dash line positionlabelled P, on gear 84 for the tip prehensile mode which is displacedfrom position P, by angle 0. With the initial three-jaw prehensile modeposition taken as the zero position and clockwise angular displacementof the finger assemblies 12 as seen in FIG. 3 taken as negative, acomparison of the resulting angular displacement of the fingerassemblies 12 is shown in TABLE I attached to the end of thisspecification.

As seen from TABLE I, finger assembly 12., is not effectively rotatedwhile assembly 12,, is rotated 60 clockwise and assembly 12, is rotated60 counterclockwise when gear 84 moves pin 90 from position P; toposition P As gear 84 rotates pin 90 from position P; to position Passembly 12,, is effectively rotated l counterclockwise, assembly 12 isrotated l80 clockwise and assembly 12,. is rotated l80 counterclockwise.As gear 84 rotates pin from position P,- to position P, assembly 12,, iseffectively rotated 180 counterclockwise; assembly 12,, is rotated 330clockwise, and assembly 12, is rotated 330 counterclockwise. Thus itwill be seen that the various prehensile patterns may be achieved bymotor 81 rotating drive shaft 82 counterclockwise from position P, toposition P, and clockwise from position P, back to position P,-.Therefore motor 81 is reversible and can stop gear 84 in any of theabove positions.

Angles a, B and 0 are determined by equation:

which produces an angle a 48, B 96 and a: The ratio of the pitchdiameter d, of drive gear 84 to each of the pitch diameters d and d ofdriven gears 85 and 86 is 1.25. To avoid locking mechanism 15, the pitchdiameter d of four-bar gear 91 is equal to three times the pitchdiameter (1,, of the driven gear 94. The distance r is always less thanone-half of the pitch diameter d of four-bar gear 91 and the distance ris always greater than one-half of the pitch diameter d, of

drive gear 84. Where distance r, r cos 60, the length L of four-bar link89 can be calculated as follows:

where t outside diameter pitch diameter of fourbar gear 91 or motor gear84. Thus, it will be seen that assembly 12,, is at the same position Pduring the three-jaw and wrap modes and at the same position P duringthe spherical and tip modes while the finger assemblies 12 and 12 changeposition for each mode.

As the pin 90 moves from position P to P and position P to P, the pin 92is moved through angle gr. While various sizes of the gears 84, 85, 86,88, 91 and 94 as well as the length of link 89 may be varied, onerepresentative size is set forth in TABLE 11 attached to the end of thisspecification.

While the positioning axes PA are illustrated parallel to each other itis to be understood that these axes may be skewed with respect to eachother without departing from the scope of the invention. Likewise, whileeach axis PA is also illustrated as parallel to its respective curlingplane CP, it is to be understood that it may be skewed with respect tothe plane CP without departing from the scope of theinvention. In thesame manner it is to be understood that the curlling axes CA of eachfinger assembly 12 may be skewed with respect to each other.

While specific embodiments of the invention have been disclosed herein,it is to be understood that full use may be made of modifications,substitutions and equivalents without departing from the scope of theinventive concept.

TABLE 1 COMPARISON OF ANGULAR DlSPLACEMEN-T OF FlNGERS Prehensile FingerFinger Finger Mode Assembly 12., Assembly 12,, Assembly 12,.

Thrce-Jaw 0 0 Wrap 0 60 +60 Spread +180 180 +l80 Tip +180 330 +330 TABLEII REPRESENTATIVE SIZES COMPONENT DlMENSlON SIZE d, 2.10" I 1.80 (I1.80" 11., Practical d 1.50" d .50" r .419" r .65" L 1.83

1 claim:

1. A multiple prehension manipulator mechanism comprising:

a plurality of finger means;

finger drive means for selectively opening and closing said finger meansto grasp; and,

positioning drive means for selectively positioning said finger means sothat at least two of said finger means are aligned along a common pathin direct opposition to each other at a first position, and are parallelto each other and laterally spaced from each other in a second position.

2. The mechanism of claim 1 wherein said positioning drive means furtherpositions said at least two finger means so that said finger means aredirected toward a common point displaced laterally from the common pathin a third position.

3. The mechanism of claim 2 wherein said positioning drive meansincludes a single motor.

4. The mechanism of claim 3 wherein each of said finger means includes afinger bendable in a single curling plane about a revolute axissubstantially normal to said curling plane and rotatable about apositioning axis substantially parallel to said curling plane.

5. The mechanism of claim 4 wherein each of said fingers has a normalsubstantially straight open position in which said finger lies along aprescribed substantially straight path in said curling planesubstantially parallel to said positioning axis.

6. The mechanism of claim 5 wherein there are three finger means andfurther including a palmar member rotatably mounting each of said fingermeans for rotation about said positioning axes, said palmar memberdefining a planar working surface and said finger means located so thatsaid fingers project over said surface from spaced points and saidpositioning axes are substantially parallel.

7. The mechanism of claim 6 wherein said finger means are positioned sothat said positioning axis of each of said finger means lies in saidcurling plane of said finger in substantial alignment with saidprescribed straight path of said finger is said open position.

8. The mechanism of claim 7 wherein said positioning means furtherincludes a four-bar linkage mechanism operatively connecting said motorwith said other of said finger means so that other finger means issubstantially parallel to and in opposition to said at least two fingermeans in said second position, and is directed toward said common pointin said third position.

9. The mechanism of claim 8 wherein said four-bar linkage mechanismincludes a drive gear having a first prescribed pitch diameter doperatively connected to said motor for selected rotation by said motor;a first driven gear having a second prescribed pitch diameter doperatively connected to one of said at least two finger means forrotating said finger means about its positioning axis and drivinglymeshing with said drive gear; a four-bar gear rotatably mounted in saidbase having a third pitch diameter d a second driven gear having afourth pitch diameter d operatively connected to said other finger meansfor rotating said other finger means about its positioning axis anddrivingly meshing with said four-bar gear; and a four-bar link pinned tosaid drive gear a prescribed distance r from its center of rotation andpinned to said four-bar gear a prescribed distance r from its center ofrotation and having a prescribed length L according to the followingrelationships:

where r outside diameter of said driven gear d

1. A multiple prehension manipulator mechanism comprising: a pluralityof finger means; finger drive means for selectively opening and closingsaid finger means to grasp; and, positioning drive means for selectivelypositioning said finger means so that at least two of said finger meansare aligned along a common path in direct opposition to each other at afirst position, and are parallel to each other and laterally spaced fromeach other in a second position.
 2. The mechanism of claim 1 whereinsaid positioning drive means further positions said at least two fingermeans so that said finger means are directed toward a common pointdisplaced laterally from the common path in a third position.
 3. Themechanism of claim 2 wherein said positioning drive means includes asingle motor.
 4. The mechanism of claim 3 wherein each of said fingermeans includes a finger bendable in a single curling plane about arevolute axis substantially normal to said curling plane and rotatableabout a positioning axis substantially parallel to said curling plane.5. The mechanism of claim 4 wherein each of said fingers has a normalsubstantially straight open position in which said finger lies along aprescribed substantially straight path in said curling planesubstantially parallel to said positioning axis.
 6. The mechanism ofclaim 5 wherein there are three finger means and further including apalmar member rotatably mounting each of said finger means for rotationabout said positioning axes, said palmar member defining a planarworking surface and said finger means located so that said fingersproject over said surface from spaced points and said positioning axesare substantially parallel.
 7. The mechanism of claim 6 wherein saidfinger means are positioned so that said positioning axis of each ofsaid finger means lies in said curling plane of said finger insubstantial alignment with said prescribed straight path of said fingeris said open position.
 8. The mechanism of claim 7 wherein saidpositioning means further includes a four-bar linkage mechanismoperatively connecting said motor with said other of said finger meansso that other finger means is substantially parallel to and inopposition to said at least two finger means in said second position,and is directed toward said common point in said third position.
 9. Themechanism of claim 8 wherein said four-bar linkage mechanism includes adrive gear having a first prescribed pitch diameter d1 operativelyconnected to said motor for selected rotation by said motor; a firstdriven gear having a second prescribed pitch diameter d2 operativelyconnected to one of said at least two finger means for rotating saidfinger means about its positioning axis and drivingly meshing with saiddrive gear; a four-bar gear rotatably mounted in said base having athird pitch diameter d5; a second driven gear having a fourth pitchdiameter d6 operatively connected to said other finger means forrotating said other finger means about its positioning axis anddrivingly meshing with said four-bar gear; and a four-bar link pinned tosaid drive gear a prescribed distance r1 from its center of rotation andpinned to said four-bar gear a prescribed distance r2 from its center ofrotation and having a prescribed length L according to the followingrelationships: d1/d2 1.25 d5 3 d6 r2 d5/2 r1 > d1/2 r1 r2 cos 60* L > or= ( ( Alpha 1/2 + Alpha 5/2 + t)2 - (r2 cos 60)2) 1/2 where t outsidediameter of said driven gear - d1.